Whole-Body MPC and Online Gait Sequence Generation for Wheeled-Legged Robots

I. INTRODUCTION

• Wheeled-legged locomotion adds wheel degrees of freedom, making simplified models of simultaneous walking and driving cumbersome to design.Appropriate gait sequences are also difficult to handcraft and matter for cost of transport.Source: §I · p. 1
• The proposed whole-body MPC captures rolling constraints while jointly modeling whole-body kinematics and single rigid body dynamics.Its model simultaneously optimizes contact forces, generalized coordinates, and velocities, including torso motion and joint kinematics.Source: §I · p. 1

1) Continuous Decision-Making:

• Decomposed-task methods split foot or wheel planning from torso planning, requiring heuristics to align their solutions; single-task methods retain the continuous decision problem as a whole.Single-task planning must still solve that problem quickly enough for online robot execution.Source: 1) Continuous Decision-Making · p. 2
• LIP models trade accuracy for fast updates, SRBD models make more accurate approximations under inertia assumptions, and centroidal dynamics can be impractical for online feedback control.Foothold heuristics can guide optimization toward sub-optimal solutions and are not intuitive for hybrid locomotion that assumes a single foothold.Source: 1) Continuous Decision-Making · p. 2

2) Discrete Decision-Making:

• Hand-tuned, time-based gait design becomes difficult when choosing contact sequences for wheeled-legged robots.Source: 2) Discrete Decision-Making · p. 2

B. Contribution

• The MPC optimizes torso and wheel motion in one task and supports online adaptation of contact and swing timings using a kinodynamic wheeled-legged model.The model treats wheels as moving ground contacts and simultaneously optimizes joint velocity and ground reaction force.Source: B. Contribution · p. 2
• The whole-body MPC captures rolling constraints and dynamics to discover complex, dynamic motions that the contribution describes as impossible for decomposed-task planning.The paper compares both approaches on the same robotic platform using prediction accuracy.Source: B. Contribution · p. 2
• The paper introduces prediction accuracy as a measure of how well a receding-horizon planner predicts the robot's future state.It uses this quantity to compare single-task and decomposed-task planning.Source: B. Contribution · p. 2
• A kinematic-leg-utility concept enables online gait sequence generation without predefined contact timings or lift-off sequences.The contribution links automatic gait discovery to aperiodic behavior and reduced cost of transport.Source: B. Contribution · pp. 2–3

II. PROBLEM FORMULATION

• The MPC optimizes control inputs over a receding horizon from the latest state measurement, applying its policy until an updated one is available.Source: §II · p. 3
• The formulation includes a time-varying running cost, terminal-state cost, and equality and inequality constraints, and relies on an SLQ approach with a feedback policy.The stated approach is based on differential dynamic programming for continuous-time systems.Source: §II · p. 3
• Figure 2 visualizes the complete locomotion controller, which is verified in challenging experiments later in the paper.Source: §II · p. 3

III. HYBRID LOCOMOTION

• Rolling-direction motion makes motion primitives and gait sequences impossible to hand-tune for hybrid locomotion.Source: §III · p. 3

A. Model Predictive Control Implementation

• The single-task MPC avoids motion primitives by optimizing over the robot’s whole-body variables.Source: A. Model Predictive Control… · p. 3

1) Modeling:

• Modeling each wheel as a moving point contact with a fixed joint position avoids added wheel constraints and does not increase optimization time compared with legged robots.The wheel can be translated into wheel inputs through contact velocity and radius.Source: 1) Modeling · p. 3

A. Model Predictive Control Implementation

• Fig. 2 shows reference trajectories converted into lift-off and touch-down sequences, then MPC joint-velocity and force plans, then torque references from tracking.Source: Figure 2 · p. 3

1) Modeling:

• The state comprises torso orientation and position, COM angular and linear velocity, and joint positions; inputs are end-effector contact forces and joint velocities.Source: 1) Modeling · p. 3

• The time-varying running cost follows external commands through a quadratic cost on state and control-input errors.Source: 2) Cost Function · pp. 3–4
• Q weights state-vector error and R weights control-input error; Q is positive semidefinite and R is positive definite.State error is x(t) − xref(t), and input error is u(t) − uref(t).Source: 2) Cost Function · p. 4
• Torso reference position and linear velocity come from an external reference trajectory, while remaining state and input variables are regularized to nominal configurations.Source: 2) Cost Function · p. 4

3) Equations of Motion:

• The model combines single rigid body dynamics and leg kinematics while treating wheels as moving ground contacts with locked rotational angles.It assumes limb-joint momentum is negligible relative to lumped COM inertia and keeps full-body inertia at a nominal joint configuration.Source: 3) Equations of Motion · p. 4
• The model’s rotation matrix projects vectors from torso coordinates to world coordinates, and T(θ) maps torso-frame angular velocities to Euler-angle derivatives.The contact position rEi(qj) depends on joint positions.Source: 3) Equations of Motion · p. 4

4) Rolling Constraint:

• For a contacting leg, velocity perpendicular to the rolling direction is constrained while velocity along rolling remains unconstrained.Forward kinematics computes the perpendicular projection, accurately estimating rolling constraints without heuristics for their direction.Source: 4) Rolling Constraint · p. 4
• In the air, a leg follows a predefined swing trajectory along the terrain normal, and its ground reaction forces are zero.Source: 4) Rolling Constraint · p. 4

B. Gait Sequence Generation

• Single-task MPC enables one parameter set across motions and adapts lift-off and touch-down timings, generating aperiodic sequences over horizon T from the reference trajectory.Source: B. Gait Sequence Generation · p. 4

1) Kinematic Leg Utility:

• A leg’s utility measures its usefulness in terms of remaining kinematic reach, with values approaching zero indicating the leg needs recovery by a swing phase.The proposed utility prioritizes kinematic capability for single-task gait adaptation, unlike impulse-generation capability in the decomposed-task approach.Source: 1) Kinematic Leg Utility · p. 4
• For non-steerable wheels, each leg’s utility is defined over an ellipse that distinguishes decay along versus lateral to the fixed rolling direction.The ellipse half-axis lengths depend on the leg’s kinematic reach.Source: 1) Kinematic Leg Utility · pp. 4–5
• The utility’s position error compares the reference torso trajectory and the recent touch-down contact position with the reference wheel end-effector position.Source: 1) Kinematic Leg Utility · pp. 4–5

2) Gait Timings Generation:

• When a leg's utility falls below a threshold, it receives a constant-duration swing phase because it is near its workspace limits.The planner computes utilities over a horizon and gives the lowest-utility legs priority for swing phases.Source: 2) Gait Timings Generation · p. 5
• A swing is added only when neighboring legs are not swinging; otherwise it is postponed, restricting gaits to pure driving, hybrid static, and hybrid trotting.Source: 2) Gait Timings Generation · p. 5

IV. EXPERIMENTAL RESULTS AND DISCUSSION

• Real-world experiments compare the planner with the sequential wheel-and-torso approach on ANYmal equipped with non-steerable, torque-controlled wheels.The authors describe this as the first same-platform comparison of single-task and decomposed-task approaches.Source: §IV · p. 5

A. Experimental Setup

• The planner, tracking controller, and state estimator run in concurrent threads on one PC, with optimization problems executed online.The robot is self-contained in computation; the setup uses an Intel i7-8850H, 2.6 GHz, hex-core 64-bit PC.Source: A. Experimental Setup · p. 5

B. Prediction Error

• Prediction error measures how well a receding-horizon planner predicts the robot's future state, addressing the difficulty of quantitatively comparing real-robot planner performance.The authors note that prior reports often give optimization time, success rate, and task difficulty without measuring performance against other algorithms.Source: B. Prediction Error · p. 5
• The error compares the predicted terminal COM position, optimized T seconds earlier, with measured COM position, and is computed only at constant reference velocities.Source: B. Prediction Error · p. 5

C. Decomposed- vs Single-Task Motion Planning

• The approaches are compared using a fixed trotting gait in terms of prediction error, dynamic model, and foothold heuristic.Source: C. Decomposed- vs Single-Task Motion… · p. 5

1) Prediction Accuracy:

• 0.061 ± 0.044 m vs 0.214±0.061 m: whole-body MPC's COM prediction error improves by approximately 71%, especially at higher commanded linear velocities and yaw rates.The comparison uses a 0.8 s horizon while hybrid trotting on flat terrain.Source: 1) Prediction Accuracy · p. 5, Figure 4 · p. 6

2) Dynamic Model:

• During dynamic motions, the ZMP leaves the support polygon, so the simplified model cannot discover the demonstrated motions; its assumptions also limit accurate modeling to coplanar contacts.The paper motivates a more accurate kinodynamic model for wheeled-legged robots.Source: 2) Dynamic Model · pp. 5–6

3) Foothold Heuristic:

• At high accelerations, constant-velocity torso assumptions make inverted-pendulum footholds diverge; feedback compensates, while MPC discovers complex hybrid behaviors without a foothold heuristic.Handcrafting a heuristic for dynamic hybrid trajectories is described as cumbersome.Source: 3) Foothold Heuristic · p. 6

D. Gait Sequence Generation

• One set of cost terms supports each gait, whereas the decomposed-task approach requires retuning them for each gait pattern.Running the gait-timing generator with the decomposed approach would require additional heuristics to interpolate between pretuned cost terms.Source: D. Gait Sequence Generation · pp. 6–7
• Up to 85% lower COT comes from reduced steps; pure driving reaches around 0.1 COT at 2 m/s, a factor of two higher than hybrid trotting.The figure shows pure driving, hybrid static gaits, and hybrid trotting during high-linear-velocity motions.Source: D. Gait Sequence Generation · p. 6, Figure 7 · p. 7

V. CONCLUSIONS

• The conclusion presents online gait-sequence adaptation as enabling ANYmal to coordinate aperiodic behavior that decreases overall mission COT.The approach makes predefined contact and swing timings obsolete.Source: §V · p. 7
• Future work will augment the blind gait-sequence generator's utility function with terrain information from exteroceptive sensors.The paper describes the reported gait sequence generation as based on kinematic leg utilities.Source: §V · p. 7